Download Ulrich bundles on abelian surfaces

PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 144, Number 11, November 2016, Pages 4609–4611
http://dx.doi.org/10.1090/proc/13091
Article electronically published on April 20, 2016
ULRICH BUNDLES ON ABELIAN SURFACES
ARNAUD BEAUVILLE
(Communicated by Lev Borisov)
Abstract. We prove that any abelian surface admits a rank 2 Ulrich bundle.
Let X ⊂ PN be a projective variety of dimension d over an algebraically closed
field. An Ulrich bundle on X is a vector bundle E on X satisfying H ∗ (X, E(−1)) =
. . . = H ∗ (X, E(−d)) = 0. This notion was introduced in [ES], where various other
characterizations are given; let us just mention that it is equivalent to say that E
admits a linear resolution as an OPN -module, or that the pushforward of E onto
Pd by a general linear projection is a trivial bundle.
In [ES] the authors ask whether every projective variety admits an Ulrich bundle.
The answer is known only in a few cases: hypersurfaces and complete intersections
[HUB], and del Pezzo surfaces [ES, Corollary 6.5]. The case of K3 surfaces is treated
in [AFO]. In this short note we show that the existence of Ulrich bundles for abelian
surfaces follows easily from Serre’s construction:
Theorem 1. Any abelian surface X ⊂ PN carries a rank 2 Ulrich bundle.
Proof. We put dim H 0 (X, OX (1)) = n. Let C be a smooth curve in |OX (1)|; we
have OC (1) ∼
= ωC and g(C) = n + 1. We choose a subset Z ⊂ C of n general points.
Then Z has the Cayley-Bacharach property on X (see for instance [HL], Theorem
5.1.1): for every p ∈ Z, any section of H 0 (X, OX (1)) vanishing on Z {p} vanishes
on Z. Indeed, the image V of the restriction map H 0 (X, OX (1)) → H 0 (C, OC (1))
has dimension n − 1, hence the only element of V vanishing on n − 1 general points
is zero; thus the only element of |OX (1)| containing Z {p} is C.
By loc. cit., there exists a rank 2 vector bundle E on X and an exact sequence
(1)
s
0 → OX −−→ E −→ IZ (1) → 0.
Let η be a non-zero element of Pico (X). Then h0 (ωC ⊗ η) = n, and so
H 0 (C, ωC ⊗ η(−Z)) = 0 since Z is general; and therefore H 0 (X, IZ η(1)) = 0. Since
χ(IZ η(1)) = 0, we have also H 1 (X, IZ η(1)) = 0; from the above exact sequence we
conclude that H ∗ (X, E ⊗ η) = 0.
The zero locus of the section s of E is Z; since det E|C = OC (1) = ωC , we get
an exact sequence
(2)
s|C
0 → OC (Z) −−−→ E|C −→ ωC (−Z) → 0.
Received by the editors December 9, 2015 and, in revised form, January 11, 2016.
2010 Mathematics Subject Classification. Primary 14J60; Secondary 13D02.
c
2016
American Mathematical Society
4609
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4610
ARNAUD BEAUVILLE
As above, the cohomology of ωC ⊗ η(−Z) and η(Z) vanishes, and hence
H ∗ (C, (E ⊗ η)|C ) = 0. Now from the exact sequence
0 → E(−1) → E → E|C → 0
we conclude that H ∗ (X, E ⊗ η(−1)) = H ∗ (X, E ⊗ η) = 0, hence F := E ⊗ η(1) is
an Ulrich bundle.
Remarks. 1) There is no Ulrich line bundle on a general abelian surface X. Indeed,
a line bundle M on X with χ(M ) = 0 satisfies c1 (M )2 = 0 by Riemann-Roch; since
X is general, we have NS(X) = Z, hence M is algebraically equivalent to OX . Thus
if L is an Ulrich line bundle, L(−1) and L(−2) must be algebraically equivalent to
OX , a contradiction.
On the other hand, some particular abelian surfaces do carry an Ulrich line
bundle. Let (A, OA (1)), (B, OB (1)) be two polarized elliptic curves, and let α, β
be non-zero elements of Pico (A) and Pico (B). Put X = A × B and OX (1) =
OA (1) OB (1). Then α(1) β(2) is an Ulrich line bundle for (X, OX (1)).
2) It follows from the exact sequence (2) that E|C is semi-stable, hence E, and
consequently F , are semi-stable (actually any Ulrich bundle is semi-stable; see
[CKM, Proposition 2.12]). Moreover if F is not stable, there is a line bundle L ⊂ E
with (L · C) = n, so that L|C must be isomorphic to OC (Z) or ωC (−Z). But we
have 2 = dim Pico (X) < dim Pico (C) = n + 1, so for Z general OC (Z) and ωC (−Z)
do not belong to the image of the restriction map Pic(X) → Pic(C). Therefore F
is stable.
3) We have constructed the vector bundle E from a curve C ∈ |OX (1)|, a subset Z
of C and an extension class in Ext1 (IZ (1), OX ). This space is dual to H 1 (X, IZ (1));
from the exact sequence 0 → IZ (1) → OX (1) → OZ (1) → 0 we get h1 (IZ (1)) =
h0 (IZ (1)) = 1, thus the extension class is unique up to a scalar. It is not difficult
to prove that H 0 (X, E) = Cs; hence E determines Z = Z(s) and the curve C, so
it depends on dim |C| + Card(Z) = 2n − 1 parameters. To get an Ulrich bundle we
put F = E ⊗ η(1) with η ∈ Pico (X); the line bundle η is determined up to 2-torsion
by det F = η 2 (3). Thus our construction depends on 2n + 1 parameters.
On the other hand, the moduli space of stable rank 2 vector bundles with the
same Chern classes as F is smooth of dimension 2n + 2 [M]; the Ulrich bundles
form a Zariski open subset MU of this moduli space. Therefore our construction
gives a hypersurface in MU .
References
[AFO] M. Aprodu, G. Farkas, A. Ortega, Minimal resolutions, Chow forms and Ulrich bundles
on K3 surfaces, preprint arXiv:1212.6248. J. Reine Angew. Math., to appear.
[CKM] Emre Coskun, Rajesh S. Kulkarni, and Yusuf Mustopa, On representations of Clifford algebras of ternary cubic forms, New trends in noncommutative algebra, Contemp. Math., vol. 562, Amer. Math. Soc., Providence, RI, 2012, pp. 91–99, DOI
10.1090/conm/562/11132. MR2905555
[HUB] J. Herzog, B. Ulrich, and J. Backelin, Linear maximal Cohen-Macaulay modules over
strict complete intersections, J. Pure Appl. Algebra 71 (1991), no. 2-3, 187–202, DOI
10.1016/0022-4049(91)90147-T. MR1117634 (92g:13011)
[ES]
David Eisenbud, Frank-Olaf Schreyer, and Jerzy Weyman, Resultants and Chow forms
via exterior syzygies, J. Amer. Math. Soc. 16 (2003), no. 3, 537–579, DOI 10.1090/S08940347-03-00423-5. MR1969204 (2004j:14067)
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ULRICH BUNDLES ON ABELIAN SURFACES
[HL]
[M]
4611
Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, 2nd
ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010.
MR2665168 (2011e:14017)
Shigeru Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3
surface, Invent. Math. 77 (1984), no. 1, 101–116, DOI 10.1007/BF01389137. MR751133
(85j:14016)
Laboratoire J.-A. Dieudonné, UMR 7351 du CNRS, Université de Nice, Parc Valrose,
F-06108 Nice cedex 2, France
E-mail address: [email protected]
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